document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=419359>Question 419359</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Logarithms using change of base formula;<BR>\n" );
document.write( "log2(19)<BR>\n" );
document.write( "The 2 is very small and on the bottom and the 19 is large.<BR>\n" );
document.write( "Can you help me to understand how to do this problem? Thanks<BR>\n" );
document.write( "Debbie </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #293476 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=293476\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If you're talking about<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%2819%29%29 ALIGN=MIDDLE  ALT=\"log%282%2C+%2819%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "then this is called \"the base 2 logarithm of 19\". It represents the exponent you would put on a 2 to get a 19:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Elog%282%2C+%2819%29%29+=+19 ALIGN=MIDDLE  ALT=\"2%5Elog%282%2C+%2819%29%29+=+19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I think your problem is to find out what number is this exponent you put on a 2 to get 19.<br><BR>\n" );
document.write( "Some logarithms are easy to find. For example <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%285%2C+%2826%29%29 ALIGN=MIDDLE  ALT=\"log%285%2C+%2826%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the exponent for 5 that results in 25. We know what exponent on a 5 results in 25. It is 2. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%285%2C+%2825%29%29+=+2 ALIGN=MIDDLE  ALT=\"log%285%2C+%2825%29%29+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "But what exponent on a 2 results in 19? <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E4+=+16 ALIGN=MIDDLE  ALT=\"2%5E4+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E5+=+32 ALIGN=MIDDLE  ALT=\"2%5E5+=+32\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the exponent that results in 18 is going to be somewhere between 4 and 5. This is about as far as we can reasonably go without the help of a calculator.<br><BR>\n" );
document.write( "But how do we use a calculator to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%2819%29%29 ALIGN=MIDDLE  ALT=\"log%282%2C+%2819%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >? There is no button for base 2 logarithms. The base of a logarithm can be any positive number (except 1). So it would seem calculators would need an infinite number of buttons to handle all the different possible logarithms.. Instead of having a button for each logarithm your calculator your calculator has buttons for logarithms of just one or two bases, 10 and/or \"e\". The base 10 log button has \"log\" on it and the base e log button has \"ln\" on it. To find logarithms of other bases you use the change of base formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29%7B%7B%7B%0D%0Ato+express+the+logarithm+you%27re+looking+for+in+terms+of+base+10+or+base+e+logarithms.+Then+you+can+use+your+calculator.+Changing+your+logarithm+your+logarithm+into+base+10+logarithms+we+get%3A%0D%0A%7B%7B%7Blog%282%2C+%2819%29%29+=+log%28%2819%29%29%2Flog%28%282%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Or we could use base e logarithms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%2819%29%29+=+ln%2819%29%2Fln%282%29 ALIGN=MIDDLE  ALT=\"log%282%2C+%2819%29%29+=+ln%2819%29%2Fln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Both of these will work and they will both result in the same answer. Just use your calculator to find the two logarithms and then divide them. </SPAN>\n" );
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